Explicit computation of Gross–Stark units over real quadratic fields

Abstract

Text. We present an effective and practical algorithm for computing Gross-Stark units over a real quadratic base field F. Our algorithm allows us to explicitly construct certain relative abelian extensions of F where these units lie, using only information from the base field. These units were recently proved to always exist within the correct extension fields of F by Dasgupta, Darmon, and Pollack, without directly producing them.Video. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=8h1-GW-sTNc.

LuCaNT: LMFDB, Computation, and Number Theory

“…The computation of Gross-Stark units over real quadratic fields was studied in [TY13] when p splits in F , and [FL22] for p inert in F . In the real-analytic setting, in [CR00] Cohen and Roblot used Stark's conjectures to compute wide Hilbert class fields of real quadratic fields, and similar algorithms form the basis for general algorithms to compute ray class fields in pari/GP.…”

On $p$-adic multiple Barnes-Euler zeta functions and the corresponding log gamma functions

“…In [33], Tangedal and Young showed that, for any real quadratic field with prime p splits completely in it, the p-adic multiple log gamma function G p,2 (x; ω 1 , ω 2 ) presents a representation for the derivative at s = 0 of the p-adic partial zeta function associated with any element in certain narrow ray class groups.…”